8. let ( f(x)=(x - 7)^{6} ). find an interval of ( x ) on which the average rate of change of ( f ) is 0.
8. let ( f(x)=(x - 7)^{6} ). find an interval of ( x ) on which the average rate of change of ( f ) is 0.
Answer
Explanation:
Step1: Recall the formula for average rate of change
The average rate of change of a function (y = f(x)) over the interval ([a,b]) is (\frac{f(b)-f(a)}{b - a}).
Step2: Simplify the function (f(x)=(x - 7)^0)
By the zero - exponent rule (a^0=1) for (a\neq0). So (f(x)=(x - 7)^0 = 1) for (x\neq7).
Step3: Calculate the average rate of change
Let (a) and (b) be two values in the domain of (f(x)) (i.e., (a\neq7) and (b\neq7)). Then (f(a)=1) and (f(b)=1). The average rate of change (\frac{f(b)-f(a)}{b - a}=\frac{1 - 1}{b - a}=0) for any interval ([a,b]) where (a\neq7) and (b\neq7). For example, take the interval ([0,1]).
Answer:
Any interval ([a,b]) where (a\neq7) and (b\neq7). For instance, the interval ([0,1]) (the answer is not unique).